X N Feng(馮夏寧) and L F Wei(韋聯(lián)福)

1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing&School of Physics and Astronomy,Sun Yat-Sen University(Zhuhai Campus),Zhuhai 519082,China

2Information Quantum Technology Laboratory,School of Information Science and Technology,Southwest Jiaotong University,Chengdu 610031,China

Keywords: quantum metrology,multi-path interferometers,quantum measurement

1. Introduction

It is well-known that interferometers (IFRs) are particularly important state-of-the-art setups in present-day precise measurements,[1]e.g.,for the atomic clocks,sensitive magnetometery, and the quantum manipulations of qubits. Indeed,beginning with the original Ramsey IFR by using a two-level atom[2]or its formally equivalent Mach–Zehnder IFR with two optical paths,[3–5]various generalized versions of IFRs have been developed for many metrologic applications.[6–8]For example, the squeezed light IFRs for gravitational wave detections,[9–11]the matter wave IFRs with cold atoms (with the shorter de Broglie wavelengths) for the precise measurements of gravity and fine structure constants,[12–15]and also the Sagnac IFRs for rotation measurements,[16]etc.

Given that the quantum many-body entanglements are very fragile and also hard to be prepared, an alternative way for improving the interferometer sensitivity by using the multipath interferometry have attracted much attention in recent years.[25–31]Physically, the resolution of the interferometric signals and thus the sensitivity of the relevant IFRs can be improved by increasing the number of interference paths. In this kind of multi-mode interferometers,the realization of the beam-splitter (BS) for the separation and recombination of the multi-mode evolutions is the key operation, which can be theoretically described by aN×Nquantum Fourier transformation (QFT) forN-mode interferometers.[26]For instance,the passive linear optical devices with uncorrelated singlephoton inputs[26–30]for optical interferometers, the Kapitza–Dirac pulse[15]for implementing the sub-shot noise limit of atomic interferometers.[32]It is shown theoretically that, the sensitivity of phase estimation can be enhanced to beat the shot-noise limit(SNL)with respective to the number of interference paths.[26]The imperfection of the multi-path optical interferometers, due to photon losses, is also investigated.[28]Specifically,the recent experiment,[33]demonstrated by a fivestate interferometer, showed the 1.75 times resolution higher than that with the usual two-state one.

2. M-level atomic Ramsey IFRs to estimate unknown relative phases

With such an experimental interference distribution,the information of the desired relative phaseθcan be extracted. Equation(3)shows that the population of the state|k/=1〉is just a phase shift 2kπ/Mcompared with state|1〉.

Fig.1. The schematic of an M-level atomic IFR,implemented by two QFTs in a projective measurement of the atom,for estimating the relative phase θ acquired by a free evolution.

We now discuss what is the uncertainty ?θMof the measured phaseθby running the above IFR afterNtrials. This can be determined by using the well-known error-propagation formula and the Cramer–Rao inequality.[34]For example, for a single projective measurement on the state|1〉,we have

theθ-independent QFI of the state|ψo(hù)ut〉. Equation (5) indicates clearly that the SQL in the original Ramsey IFRs can be really enhanced by running theM-level atomic IFRs forM ≥3.

which is related to both the numberMof the atomic levels and theθwith a period of 2π. This implies that,by properly

Fig.2. Attainability /?θM (of Eq.(4))of the sensitivity of the phase measurements versus the number M of the atomic levels and the estimated phase θ by a single projective measurement of ground state|1〉.

The above observations indicate that, although theMlevel atomic IFRs proposed here could be utilized to improve the sensitivity of the phase estimation,the region(within a 2πperiod)optimal is limited for a single projective measurement on a certain state. Fortunately, most of the phases estimated with the practical IFR applications, e.g., the sensitivity magnetometery, gravitater and fine-structure constants, are sufficiently small,and thus the present IFRs should be robustness.

In fact, the tradeoff between the sensitivity and effective range can be overcome by performing a POVM{|k〉〈k|}instead of a single projective measurement. This is because every projective measurement covers a separate effective regime,the sum in measurements just covers the whole regime of[0,2π]as shown in Fig.3. Actually, this can be viewed from the perspective of Fisher information with respective to the POVM of projective measurement. This is verified numerically in Fig.4.

It is worth emphasizing that the above assumption of equivalent energy splittings to simplify the presentation of our generic proposal is practically unnecessary. In fact, for the multilevel atoms with inequivalent energy splittings, certain additional quantum operations,typically such as the spinecho-like ones, can be added to refocus the unwanted additional dynamical phases. This implies that the proposed IFRs should still work,even for the generic multilevel atoms.

3. Improving the sensitivity of the magnetometery with the IFRs using three-level NV centers in diamond

Single NV centers in diamond are perfect quantum objects to implement quantum coherent manipulations, as they possess significantly long coherence times and can be operated at room temperature. In fact, a series of experiments have been recently demonstrated to implement the IFRs using single NV centers with two Zeeman levels for magnetic sensors at nanoscale space resolutions,see,e.g.,Refs.[35,36]and references therein. Typically, a 3 nT magnetic field has been detected at the sensitivity of 0.5 μT·Hz?1/2for a diamond nanocrystal with a diameter of 30 nm.[37]However,the further improvements of the sensitivity with the standard IFRs would be principally limited by the SQL bound.

By just generalizing the IFRs with two-level NV centers to those with three-level ones, we now specifically demonstrate that the original two-level SQL can be enhanced. Without loss of generality, we consider the simplest V-type threelevel configuration of the NV center shown in Fig.5,whereinDgs?ω21=ω31?Dgsandωjk=ωj ?ωk,j>k. Here,Dgsis the ground triplet-state zero-field splitting,i.e.,the Zeeman splitting ofS=1, which is proportional to the projection of the applied magnetic field along the NV symmetry axis.

Fig.5. The lower three Zeeman levels and their transition relations of an NV center in diamond. Here,g denotes g-factor,μB is the Bohr magneton,νi with i=1,2 is the frequency of the allowed transition,and B is supposed as a given applied magnetic field to obtain the desired frequencies ω2 and ω3, while ?B is the weak magnetic field expected to be measured.

As mentioned above,the key of our phase estimation algorithm with the proposedM-level IFRs is to implement the relevant QFT operations. For the present three-level system,the QFT reads[38]The evolutionsUk=1,2,3,Dto implement the required QFT can be successively achieved by applying the relevant microwave pluses designed accordingly.

First, to realize theU1-evolution using the transition|1〉?|2〉,which is practically dipole forbidden,we apply two microwave pulses with the frequenciesν1andν2,respectively.These pulses drive the system by the following Hamiltonian:

whereΛ2=diag(eib1, eib2, eib3),andb1=a1,b2=a2?ν1t2,b3=a3+β2t2forθ2+a1?a2=?π/6. Thus,after these operations,the evolution(iΛ2)U2U1is implemented.

Fig.6. The populations of states|1〉,|2〉and|3〉as a function of phase θ.

Fig.7.The sensitivities characterized by the error-propagation formula,k (dashed)and the Fisher information(red solid),,corresponding to the projective measurements POVM(P1,2,3).

With the phaseθ=δT0determined, the detuningδcan be derived such that the magnetic field ?B=ˉhδ/(gμB)can be estimated, withμBbeing the Bohr magneton andgdenoting theg-factor.Specifically,for a weak magnetic field required to be estimated,e.g.,Bz ～lnT,[35,37]and for the typical free evolution durationt ～0.3 ms,Fig.8 shows clearly that the reachable sensitivity of the IFRs with the present three-level NVcenters really beats the optimal bound in the original atomic IFRs using the usual two-level ones.

Fig. 8. Reachable sensitivity ?Bmin vs the measured magnetic field strength B,implemented by the IFRs with single two-levels NV centers(red solid line) which is given as ?Bmin = hˉ/(gμBT) with T and[37]T2 denoting the measurement and coherent time, respectively,and three-level(blue dash-dot line)NV centers,respectively. Here,the free evolution time T0 is typically set as π/10 ms.[35]It is seen that reachable sensitivity with the three-level atomic IFRs is higher than that with the usual two-level atomic IFRs for the applied weak magnetic field, i.e.,B<40 nT.

Note that, the sensitivity of the above phase estimation can also be enhanced by performing the IFRs with only two spin basis{?1,+1}(states|2〉,|3〉) for phase integration,which is called the double quantum (DQ) coherence magnetometry(Refs.[39,40]),since it accumulates phase at twice the rate of traditional single quantum(SQ)coherence magnetometry with states|1〉and|2〉(or|3〉).Though DQ magnetometry provides enhanced susceptibility to target magnetic-field signals while also making the spin coherence as twice as being sensitive to magnetic noise, making the spin coherence time for|2〉?|3〉is usually shorter than|1〉?{|2〉,|3〉}.[41]Many schemes have been put forward to prolong the coherent time by suppressing the dephasing,including the dynamical decoupling and quantum teleportation.[41–43]

4. Discussion and conclusion

Probably, one of the most challenges to experimentally demonstrates the proposed multilevel atomic IFRs such that the implementations of the required QFTs are relatively complicated and their fidelities are certainly limited by various operational imperfections(due to the inevitable noises). This is one of the common obstacles for the current implementations of all the desired quantum coherence manipulations. In this work,we have shown how to implement a QFT with pulse sequence in aV-type three-level atom with the known energy splittings. In practical cases, the external field (to be estimated)may exist during the whole process and can always not be turned off before and after the phase integration, and thus the implementations of the QFT may be imperfect. Specifically,we suppose that the real frequency of states|2〉and|3〉is symmetrically shift by a relative small detuningδasω2?δandω3+δ, due to the Zeeman effect of a external magnetic field.Under this assumption,the Hamiltonians for implementing the evolutionU1toUDshould be modified. Firstly, the Hamiltonian ～H1in Eq.(11)would be changed to

Fig. 9. The dependence of population of state |1〉 on detuning δ with different Rabi-freque} ncy ?.The distance is defnied as maxkl|(where and denote the perfect and the imperfect QFT with the detuning δ. The frequencies of states |1,2,3〉are supposed as ω1=0,ω2=0.96 GHz,ω3=1.92 GHz.

Fig. 10. The measurement population of state |1〉 as a function of phase θ under different detuning δ. Here Rabi frequencies are all set as 2π×106 with the large detuning ?=0.1 GHz.

Figure 9 shows the dependence of the distance between the theoretical imperfect QFT matrixUdQFTand the true QFT matrixU3QFTwith and without the detuningδ. The distance is defined as the maximum absolute value of the matrix elements|(UdQFT?U3QFT)kl|/|U3QFT)kl|. It is seen that the distance depends not only on the detuningδbut also on the Rabi frequency?. Ifδis significantly small,a small error in the order ofδ/?exists. To further check the validity of the QFT with the detuning, we also plot the dependence of the populations of the ground state|1〉on the phaseθf(wàn)or different detunings in Fig.10. One can see that a very small difference(less than 10?3) occurs, even forδ=10 kHz. From the above analyses,we can draw a conclusion that the proposed interferometry works even the level-spacing is not known exactly.

In conclusion,via replacing the Hadmard gates by a pair of QFTs we generalize the original Ramsey IFR with a twolevel atom to a new IFR with a multilevel atom. The sensitivity of the phase estimations with the proposedM-level IFRs could be enhanced by a factor of (M2?1)/3 compared with the usual two-level cases. Specifically, our generic proposal has been demonstrated with single three-level NV centers in diamond to improve the sensitivity of the magnetometery.Furthermore,considering the practical application,we also investigate the effect of imperfection of the evolutions on the phase estimation sensitivity by considering the impact of a small detuning,caused by an external field,on the realization of QFT.Anyway,given the IFRs proposed here using only single atoms without any inter-atom quantum entanglement and possessing the significant improvement of sensitive bound,its experimental realization and application should be valuable and interesting.